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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m

Generation of two-mode entangled states in a four-level atomic system via the Raman process Hong-rong Li a,b,⁎, Li-gang Wang b,c, Gao-xiang Li b,d, Fu-li Li a, Shi-yao Zhu e a

Non-equilibrium Condensed Matter and Quantum Engineering Laboratory, and Department of Applied Physics, Xian Jiaotong University, Xian 710049, China Institute of Theoretical Physics, The Chinese University of Hong Kong, Hong Kong, China Department of Physics, Zhejiang University, Hangzhou 310027, China d Department of Physics, Huazhong Normal University, Wuhan 430079, China e Department of Physics, Hong Kong Baptist University, Hong Kong, China b c

a r t i c l e

i n f o

Article history: Received 3 May 2010 Received in revised form 23 July 2010 Accepted 23 July 2010

a b s t r a c t In this paper, we have theoretically investigated the generation of two-mode entangled states from a fourlevel atomic system via the Raman process. We show that the degree of entanglement between the two cavity modes could be strongly adjusted by both the Rabi frequencies and the detunings of the pumping ﬁelds. Our numerical results reveal that entanglement between the steady state of the two cavity modes depends on the difference of the two detunings of the atomic levels with the classical laser ﬁelds or the difference of the two Rabi frequencies. Finally, our result also shows that when such atomic system is operated above the threshold, it is possible to obtain the macroscopic entangled states. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Gaussian entangled state is one of the key physical resources in continuous variable quantum information and quantum computation processes [1]. The degree of entanglement for an entangled resource is related directly to the success of quantum manipulation process [2]. There are many theoretical or experimental schemes for generating Gaussian entangled states, such as cold atomic ensembles or single atom with cavity QED [3–5], ion cooling and trapping [6,7], and an atomic cloud in an optical cavity [8]. Particularly, the optical nondegenerate parametric down conversion (ONPDC), which has been long envisaged as a source of two-mode squeezed states, is proved to be an effective way to produce two-mode Gaussian entangled sources. The experimental observation of the Gaussian entanglement was realized in the ONPDC with nonlinear optical processes and linear elements [9]. Recently, Xiong et al. [10] proposed a novel ONPDC scheme based on the two-mode correlated spontaneous emission process and concluded that the creation of two-mode entanglement with photon numbers up to tens of thousands seems achievable even in the system with the presence of cavity loss. Tan et al. [11] found that the atomic collectivity can effectively increase the strength of the ONPDC process in the three-level cascade atoms and lead to the signiﬁcant enhancement of the entanglement between the two cavity modes. Li et al. [12] proposed a scheme to effectively improve the strength of the ONPDC with the help of the auxiliary ⁎ Corresponding author. Department of Applied Physics, Xian Jiaotong University, Xian 710049, China. E-mail address: [email protected] (H. Li). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.07.069

atomic transition. Kiffner et al. [13] investigated a single four-level atom trapped in a doubly resonant cavity, and they found that the macroscopic entangled light can be produced even when the system works in the weak-coupling regime. The entanglement of a two-mode Gaussian state has been well characterized [14] by its covariance matrix (CM), which can be determined experimentally by a Single Homodont Detector [15]. Based on the physical description of the entanglement, it is convenient for us to investigate how to generate a high degree of entanglement of the Gaussian states via pumped atomic systems under different control parameters [10–13,16]. In this paper, we consider the generation of two-mode entangled states from a fourlevel atomic system via the Raman process. The atoms are trapped in a ring doubly resonant cavity and pumped with two controlling laser ﬁelds. The similar schemes were investigated experimentally as the single atom laser [16], and as a macroscopic entangled light of singleatom [13], however, in our scheme for the pumping ﬁelds we mainly focus on the Raman process with large detunings. As we know, the Raman process has several advantages such as excellent signal-tonoise ratio and wide tunability compared with the conventional nearresonant schemes. In our scheme, all the detunings between the atomic levels and the controlling ﬁelds can be ﬂexibly adjusted. We directly calculate entanglement degree of the two cavity modes of the system, and our work determines entanglement of the system with a wide range of control parameters. This paper is organized as follows. In Section 2, we present the atomic system and derived the master equation. In Section 3, we introduce the entanglement measure of two-mode Gaussian states. Section 4 is devoted to discuss how the external driving ﬁelds adjust

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or amplify the degree of entanglement for the generated two-mode states via the Raman process. At last, we summarize our result in Section 5.

where Lspon ρ =

2. Theoretical model and derivation of the master equations We consider a cloud of four-level atomic system embedded in a ring doubly resonant cavity, whose conﬁguration as shown in Fig. 1. The atomic system interacts with the two cavity modes as well as two classical laser ﬁelds. The two pumped laser ﬁelds have different frequencies and different Rabi intensities, the ﬁrst laser ﬁeld with frequency ωa and Rabi frequency Ωa couples to the |1〉 → |3〉 transition, and the second ﬁeld with frequency ωb and Rabi frequency Ωb drives the |2〉 → |4〉 transition. The two cavity modes with frequencies ν1 and ν2 couple to the atomic transitions |1〉 → |4〉, and |2〉 → |3〉 with coupling constants g1 and g2, respectively. In the rotating-wave approximation, the interaction Hamiltonian for the system in the interaction picture is HI = HI1 + HI2 + HI3 ;

ð1Þ

HI1 = −Δa S11 −Ωa exp ð−iϕ1 ÞS13 −Ωa exp ðiϕ1 ÞS31 ;

ð1aÞ

HI2 = −Δb S22 −Ωb exp ð−iϕ2 ÞS24 −Ωb exp ðiϕ2 ÞS42 ;

ð1bÞ

þ

þ

HI3 = V14 S14 + V14 S41 + V23 S23 + V23 S32 ;

ð1cÞ

Sji = jjihi j; V14 = g1 expð−iΔ1 t Þa1 ;

n o ρ˙ f = −itratom ½HI ; ρ + tratom Lspon ρ + Lcav ρ = −ið½V14 ; ρ41 + ½V41 ; ρ14 + ½V23 ; ρ32 + ½V32 ; ρ23 Þ n o + tratom Lspon ρ + Lcav ρ :

ð5Þ

To evaluate the atomic matrix elements ρij ði; j = 1; 2; 3; 4Þ, we assume that the two classical laser ﬁelds are taken into account to all orders in the Rabi frequencies Ωa and Ωb and the two cavity modes are only treated to ﬁrst order in the coupling constants gi ði = 1; 2Þ. For simplicity, we also assume that all the atomic decays are the same, i.e. γi = γ. Then the equations of motion for the atomic transitions ρ14 and ρ32 are ρ˙ 14 = −ðγ−iΔa Þρ14 −i½−Ωa expð−iϕ1 Þρ34 + Ωb exp ð−iϕ2 Þρ12 + V14 ρ44 −ρ11 V14 ; ð6Þ

ð2Þ

ρ˙ 32 = −ðγ + iΔb Þρ32 −i½−Ωa expðiϕ1 Þρ12 + Ωb exp ðiϕ2 Þρ34 + V32 ρ22 −ρ33 V32 :

Here aj ðj = 1; 2Þ are the annihilation operators of the two cavity modes, ϕ1 and ϕ2 are the phases of the laser ﬁelds, Δa and Δb are the detunings of two laser ﬁelds to the corresponding atomic transitions, Δ1 = δ1 − Δa, and Δ2 = δ1 − Δb with δ1 and δ2 are the detunings of the two cavity modes to the corresponding atomic transitions. The master equation for the combined system of the atom and the two cavity modes is ρ˙ = −i½HI ; ρ + Lspon ρ + Lcav ρ;

ð4Þ

is the term for spontaneous decays, and the parameters γi are the decay rates between different transitions, and Lcav ρ = þ þ ∑i = 1;2 κi 2ai ρaþ i −ai ai ρ−ρai ai accounts for decay of the two cavity modes. The master equation for the reduced density operator of the cavity modes ρf is obtained by taking a trace over atoms, which leads to [17]

where

V23 = g2 expð−iΔ2 t Þa2 :

1 fγ ½S ; ρS13 + γ2 ½S41 ; ρS14 2 1 31 + γ3 ½S32 ; ρS23 + γ4 ½S42 ; ρS24 + H:c:g

ð3Þ

ð7Þ

ρ˙ 34 = −i½−Ωa expðiϕ1 Þρ14 + Ωb exp ð−iϕ2 Þρ32 + V32 ρ24 −ρ31 V14 ; ð8Þ ρ˙ 12 = −½γ−iðΔa −Δb Þρ12 −i½−Ωa expð−iϕ1 Þρ32 + Ωb exp ðiϕ2 Þρ14 + V14 ρ42 −ρ13 V32 : ð9Þ With the conditions Ωi N N gj, ði = a; b and j = 1; 2Þ, we have the following equations ρ˙ 11 = −2γρ11 + iΩa ½expð−iϕ1 Þρ31 − exp ðiϕ1 Þρ13 ; ρ˙ 22 = −2γρ22 + iΩb ½expð−iϕ2 Þρ42 − exp ðiϕ2 Þρ24 ; ρ˙ 33 = γðρ11 + ρ22 Þ + iΩa ½expðiϕ1 Þρ13 − exp ð−iϕ1 Þρ31 ;

ð10Þ

ρ˙ 44 = γðρ11 + ρ22 Þ + iΩb ½expðiϕ2 Þρ24 − exp ð−iϕ2 Þρ42 ; ρ˙ 13 = −ðγ−iΔa Þ + iΩa expð−iϕ1 Þðρ33 −ρ11 Þ; ρ˙ 24 = −ðγ−iΔb Þ + iΩb expð−iϕ2 Þðρ44 −ρ22 Þ: Let ρ˙ ij = 0 in Eq. (10), under the steady limit, we could obtain ρij = zij ρf ;

ð11Þ

where Fig. 1. Conﬁguration of the system. A cloud of four-level atomic system is embedded in a ring doubly resonant cavity. The transitions |1〉 → |3〉 and |2〉 → |4〉 are driven by two laser ﬁelds with Rabi frequencies Ωa and Ωb, respectively. The two cavity modes with frequencies ν1 and ν2 couple to the atomic transitions |1〉 → |4〉, and |2〉 → |3〉 with coupling constants g1 and g2, respectively.

z11 z33 z44 z24 z13

= = = = =

z22 = z1 z2 = z; ð1 + z1 Þz2 = z; ð1 + z2 Þz1 = z; −iV24 = ½zðγ−iΔb Þ; −iV13 = ½zðγ−iΔa Þ;

ð12Þ

H. Li et al. / Optics Communications 283 (2010) 5269–5274

and 2 2 2 z1 = jV13 j = γ + Δa ; 2 2 2 z2 = jV24 j = γ + Δb ;

ð13Þ

Then Eqs. (6)–(9) are expressed as Y˙ = −LY−iK expð−iΔt Þ;

ð14Þ iT þ where Y = ½ρ14 ; ρ32 ; ρ34 ; ρ12 T , with K = K0 a1 ρf ; ρf a1 ; aþ , 2 ρ; ρf a2 and h

γ−iΔa 6 6 0 L=6 6 iV 4 13

0

iV13

−iV24

γ + iΔb

−iV24

iV13

−iV24

0

0

iV13

0

γ−iðΔa −Δb Þ

−iV24

3 7 7 7; 7 5

ð15Þ

6 6 K0 = 6 4

g1 z44

−g1 z11

0

0

0

g2 z24

0

−g1 z31

g2 z24

g1 z42

0

0

0

3

−g2 z33 7 7 7: 0 5

ð16Þ

−g2 z13

Here we assume that the condition of two-photon resonance is satisﬁed, i.e. Δ1 = −Δ2 = −Δ; or ωa + ωb = ν1 + ν2 :

ð17Þ

The steady solution of Eq. (14) is Y = −iXK exp ðiΔt Þ;

ð18Þ

where X = ðL + iΔIÞ−1 , and I is a 4 × 4 unit matrix. Finally, we obtain the master equation for density matrix ρf of the cavity modes þ þ + g1 μ12 aþ ρ˙ f = g1 μ11 aþ 1 a1 ρf −a1 ρf a1 1 ρf a1 −ρf a1 a1 þ þ þ + g2 μ33 a2 aþ 2 ρf −a2 ρf a2 + g2 μ34 a2 ρf a2 −ρf a2 a2 +

þ þ þ ðg1 μ13 −g2 μ32 Þaþ 1 a2 ρf −ðg1 μ14 −g2 μ31 Þρf a1 a2

+ ðg1 μ14 +

þ g2 μ32 Þaþ 1 ρf a2 −ðg1 μ13

+

ð20Þ

The partial transposed matrix of the covariance matrix ΓG has the smallest symplectic eigenvalues [14] ED E 2 D 1 þ þ 2 c− = ∑ ai ai ai ai + 1 −2jha1 a2 ij + 2 i=1 ﬃ þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ 2 þ − a1 a1 + a2 a2 + 1 aþ + 4jha1 a2 ij2 : 1 a1 −a2 a2

ð22Þ

Separability or entanglement of a quantum state may quantitatively be measured. Several quantitative measurements of quantum entanglement have been proposed [20]. Vidal and Werner [21] suggested a quantitative evaluation of entanglement by the use of the modulus of the sum of the negative eigenvalues of the partial transposed deﬁned as density matrix, the so-called pnegativity, ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ NðρÞ = ρT1 −1 = 2, where jjσ jj = tr σ þ σ represents the 1

1

trace norm of the operator σ. Very closely related to the negativity, another entanglement measure is the logarithmic negativity EN ðρÞ = log2 ρT1 1 . In terms of the smallest symplectic eigenvalue (Eq. (22)), the logarithmic negativity of TAGS with the CM can be represented in the form [21] EN ðρÞ = maxf0; − log2 2c− g:

ð23Þ

With the above notation, the master equation (Eq. (19)) is rewritten as ξ˙ = Mξ + η;

ð19Þ

þ g2 μ31 Þaþ 2 ρf a 1

+ H:c: + Lcav ρf ; where μ = − XK0. 3. Entanglement degree of Gaussian states Because the terms of Eq. (19) are all quadratic terms of the Bosonic operators aj and aþ j ðj = 1; 2Þ, the states of the cavity ﬁelds governed by Eq. (19) in phase space should be the two-mode Gaussian states (TAGS) if the initial states are also Gaussian. The entanglement properties of TAGS are completely determined by the covariance A G matrix (CM) ΓG = [18,19]. For the system we considered Gþ B here, if we assume that the initial state of the two-mode cavity modes is state, all the nonzero elements of ΓG are A11 = A22 = þvacuum a1 a1 + 1 = 2, B11 = B22 = aþ a n d G12 = G21 = 2 a2 + 1 = 2, −ha1 a2 i. Simon [19] showed that the Heisenberg uncertainty relation 1 applied to the matrix ΓG equates to the restriction ΣðΓG Þ≤ + 4 4det ðΓG Þ; where ΣðΓG Þ = detA + detB + 2detG. The necessary and sufﬁcient condition for separability of TAGS is the positivity of the partial transposed density matrix (PPT criterion)

ð21Þ

The evidence of entanglement of TAGS can be rewritten as the very simple inequality 1 c− b : 2

and 2

[19]. The partial transpose operation on the Gaussian states characterized by the CM corresponds to ﬂipping the sign of detG. Let ˜ ðΓG Þ = detA + detB−2detG. The PPT criterion can be represented in Σ the form [14,19] ˜ ðΓ Þ ≤ 1 + 4det ðΓ Þ: Σ G G 4

z = z1 + z2 + 4z1 z2 :

2

5271

where ξ =

ð24Þ

þ þ þ T , and aþ 1 a1 ; a2 a2 ; ha1 a2 i; a1 a2

M11 = g1 ðμ 11 + μ 11 + μ 12 + μ 12 Þ−2κ1 ;

M22 = −g2 ðμ34 + μ 34 + μ 33 + μ 33 Þ−2κ2 ;

M33 = g1 ðμ 11 + μ 12 Þ−g2 ðμ 33 + μ 34 Þ −κ1 −κ2 ;

M44 = g1 ðμ 11 + μ 12 Þ −g2 ðμ 33 + μ 34 Þ−κ1 −κ2 ;

M13 = M42 = g1 ðμ 13 + μ 14 Þ ;

ð25Þ

M14 = M32 = g1 ðμ 13 + μ 14 Þ; M23 = M41 = −g2 ðμ 31 + μ 32 Þ;

M14 = M31 = −g2 ðμ 31 + μ 32 Þ : η11 = η13 =

g1 ðμ 12 + μ 12 Þ; η12 = −g2 ðμ 33 + μ 33 Þ; g1 μ 13 −g2 μ 32 ; η14 = g1 μ 13 −g2 μ 32 :

It is easily found that when the system obeys the condition that all of the real parts of eigenvalues of matrix M are negative, i.e., that is operated below the threshold, the system can approach its steady state in the long-time limit [12]. The steady solution of Eq. (24) is ξðt→∞Þ = M

−1

η:

ð26Þ

A recent work shows that all elements of CM for TAGSs can be determined experimentally [15]. In the next section, we will use the

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logarithmic negativity to measure quantum entanglement of the cavity modes.

If the condition Ωb NN g˜ 1 ; g2 is satisﬁed, the effective Hamiltonian in the dressed-state picture is obtained

4. Entanglement degree of the cavity modes

2 þ þ þ Veff = ðP3 −Pd Þ a1 a2 + a1 a2 + ∑ Pi 2ai ai + 1 ;

33

i=1

The system considered here may produce entanglement of the two cavity modes through two Raman two-photon emission processes. If the larger detuning conditions are satisﬁed, the population of the four levels of the atom reduce to z11 ≈ z22 ≈ 0; z33 ≈ Δ2a Ω2b = Δ2a Ω2b + Δ2b Ω2a ; z44 ≈ Δ2b Ω2a = Δ2a Ω2b + Δ2b Ω2a :

ð27Þ

This means the atom is always at the two lower levels |3〉 and |4〉. In order to show the physics of the process of entanglement generation, we analyze the system in the dressed-state representation of Hamiltonian HI1 and HI2. The dressed states of HI1 are j +i = sinφ1 expðiϕ1 Þ j1i−cosφ1 j3i;

where Pi is determined by sums of the projectors Ωa Ωb g˜ 1 g2 j 3ih3 j ; Ω2b + Δ2 Δa Ω g˜ g 1 1 j+ih+j + j −ih−j ; Pd = a 1 2 Ωb + Δ Ωb −Δ 2Δa Ω2a Ωb g˜ 1 Ω2a g˜ 1 1 1 j +ih + j − j −ih−j ; P1 = 2 j 3ih3 j − Ωb −Δ 2Δa Ωb + Δ Ωb + Δ2 Δa 2 2 Ω g g 1 1 P2 = 2 b 2 2 j3ih3 j− 2 j +ih+ j − j −ih−j : Ωb −Δ 2 Ωb + Δ Ωb + Δ P3 =

ð34Þ The closed two-photon transition is occurred between the states | ± 〉 ↔ |3〉 ↔ | ± 〉.

ð28Þ

j −i = cosφ1 expðiϕ1 Þj1i + sinφ1 j3i; and the dressed states of HI2 are

jþ′ i = sinφ2 expðiϕ2 Þj2i−cosφ2 j4i; j−′ i = cosφ2 expðiϕ2 Þj2i + sinφ2 j4i;

ð29Þ

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ˜ ˜ Ωi − Δ Ωi + Δ i i ˜ 2 = Δb ; ˜ 1 = Δa , Δ ; sinφi = ; and Δ where cosφi = 2Ωi 2Ωi qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ω1 = Δ2a +4jΩa j2 ; Ω2 = Δ2b + 4jΩb j2 . The interaction Hamiltonian in the dressed-state picture is then h i ′ ′ a1 fsinφ1 sinφ2 exp i λþ −λ− t j +i − h i h i ′ ′ ′ ′ −cosφ1 cosφ2 exp i λ− −λþ t j−i þ −sinφ1 cosφ2 exp i λþ −λþ t j+i þ h i ′ ′ + cosφ1 sinφ2 exp i λ− −λ− t j−i − h i −iϕ −iΔt ′ ′ + g2 e 2 a2 fsinφ1 sinφ2 exp i λþ −λ− t þ h−j h i h i ′ ′ ′ ′ −cosφ1 cosφ2 exp i λ− −λþ t − h+ j −sinφ2 cosφ1 exp i λþ −λþ t þ h+ j h i ′ ′ + cosφ2 sinφ1 exp i λ− −λ− t − h− j + H:c: −iϕ1 + iΔt

VdI ðt Þ = g1 e

h j h jg

j i j i

h j

j i

g

h j

j i

ð30Þ The effective Hamiltonian is obtained by Veff = − i VdI ðt Þ∫VdI ðτÞdτ, and below, we investigate the process of entanglement generation by considering three cases. If condition Δ2aΩ2b N N Δ2bΩ2a is satisﬁed, i.e., z33 ≈ 1, z44 ≈ 0, the atom is approximately a three-level system and the level |1〉 is adiabatic canceled, the effective Hamiltonian is Heff ðt Þ = −Ωb j 2ih4 j + g˜ 1 a1 expðiΔt Þj 3ih4j + g2 a2 expð−iΔt Þ j 2ih3 j + H:c:;

ð31Þ where g˜ 1 = g1 Ωa = Δa and we set ϕ1 = ϕ2 = 0. As the Hamiltonian, the transition |3〉 ↔ |1〉 ↔ |4〉 can be effectively estimated as a twophoton transition between levels |3〉 ↔ |4〉. We express the effective Hamiltonian in the above dressed-state picture and then cosφ1 ≈0; pﬃﬃﬃ sinφ1 ≈1; cosφ2 ≈sinφ2 ≈1 = 2, the interaction Hamiltonian is thus g˜ iΔt iΩ t −iΩ t H ′ ðt Þ = p1ﬃﬃﬃ a1 e j3ih+je b −j3ih−je b 2 g −iΔt −iΩ t iΩ t + p2ﬃﬃﬃ a2 e j+ih3je b + j−ih3 je b + H:c:; 2

ð32Þ

Fig. 2. The logarithmic negativity En of two cavity modes versus the two detunings Δa and Δb. The parameters are Ωb = 20g, Δ = − 10g, g1 = g2 = g, γ = g, κ1 = κ2 = 10 − 2g, ϕ1 + ϕ2 = π /2, and with a) Ωa = 25g, b) Ωa = 65g.

H. Li et al. / Optics Communications 283 (2010) 5269–5274

5273

Symmetrically, if condition Δ2bΩ2a N N Δ2aΩ2b is satisﬁed, i.e., z44≈1, z33≈0, the level |2〉 is adiabatic canceled. The transition is taken | ± ′〉 ↔ |4〉 ↔ | ± ′〉 via another two-photon emission process. To consider entanglement of the system under symmetric case, we set Ω a = Ω b = Ω, Δ a = Δ b = Δ 0 , ϕ 1 = ϕ 2 = ϕ, Δ = 0, k 1 = k 2 = k, g1 = g2 = g, φ1 = φ2 = φ and the effective Hamiltonian is then h g 2 cos2 φsin2 φe2iϕ a1 a2 ð j+ih+j− j−ih− j Þ Ω i + ðj+′〉〈+′j−j−′〉〈−′jÞ

Veff −sym = −

2

+

g þ 4 4 a a ðcos φj+ih+ j−sin φj−ih− jÞ Ω 1 1

+

g2 þ 4 4 a a ðcos φj+′〉〈+′j−sin φj−′〉〈−′jÞ + H:c: Ω 2 2

ð35Þ

Because the population of states | + 〉 and | − 〉 are almost equal (the same for | + ′〉 and | − ′〉), there is tiny entanglement under the symmetric case. We may also check the conclusion using separable condition for two-mode states. We rewrite condition (22) as T −1 −1 Eλ = M η U M η b0:

ð36Þ

The condition is also a necessary and sufﬁcient condition to indicate entanglement, and where the nonzero elements of matrix U are U12 = U21 = − U34 = − U43 = 1/2 and ﬁnd that Eλ has the same sign with the term 4k2 Δ40 + 16k2 jΩj4 + 16k2 g 2 jΩj2 + 8k2 g 2 Δ20 + 4g 4 k2 −g 4 Δ20 −g 6 Þ, where we set γ = g. It is easy to verify that Eλ is always larger than zero when the system operates below the threshold. Thus, we conclude again that the system under symmetric case is not entangled. Not restricted by these special conditions, emission from atom to the cavity modes may include both of the two Raman two-photon emission processes, which may interfere destructively with or interfere with the enhancement of each other. To determine entanglement of the two cavity modes generally, we ﬁrst numerically consider the steady case that is given by Eq. (26). In Fig. 2, we plot the logarithmic negativity En of the two cavity modes versus the two detunings Δa and Δb for the given Rabi frequencies and other parameters. Fig. 2a and b is plotted under the condition that jΩa −Ωb jbbminfΩa ; Ωb g, and jΩa −Ωb j NN minfΩa ; Ωb g. As shown in the color bar, in the graph the hot color represents larger logarithmic negativity, and the cold color represents lower logarithmic negativity. In Fig. 2a, we set Ωa = 25g, Ωb = 20g, Δ = − 10g, g1 = g2 = g, γ = g, κ1 = κ2 = 10 − 2g, ϕ1 + ϕ2 = π /2. The non-entangled and nearly non-entangled areas, i.e. En≈ 0, are along the two diagonal area that is characterized by the condition of jΔa j≈jΔb j. The areas with little degree of logarithmic negativity, i. e. En b 0.3 is on the four cross area that corresponds to the condition of jΔa j and jΔb j are both large. The area with larger logarithmic negativity, i.e. En N 0.3 locates at the remaining area, is shown in the hot color area of the graph and approximately corresponds to the condition of jΔa −Δb j NN 0. In Fig. 2b, we set Ωa = 65g, and other parameters are the same as in Fig. 2a. We ﬁnd in Fig. 2b that the area of the non-entangled and the nearly non-entangled is smaller than that in Fig. 3a i.e. most areas in Fig. 3b are entangled, and the condition for En0 is not characterized by the condition of jΔa j≈jΔb j again except when jΔa j≈jΔb j≈0. The hot color area is different with Fig. 2b. In Fig. 2b this entangled area with a large degree of logarithmic negativity mainly locates at jΔb j∈½0; 50g and jΔb j∈½100g; 200g with jΔa j N 200g: In Fig. 3a and b, we plot the graph of En versus Ωa and Ωb. In Fig. 3a, we set Δa = 55g, Δb = 60g, and other parameters are the same as in Fig. 2. When the condition of Ωa N 70g, Ωb N 60g, and jΩa −Ωb jb10g is satisﬁed, there has been a wide range of non-entangled area. The hot color area exists in the left-middle area in the graph. Different from

Fig. 3. The logarithmic negativity En of two cavity modes versus the two Rabi frequencies Ωa and Ωb. The parameters are Δb = 60g, Δ = − 10g, g1 = g2 = g, γ = g, κ1 = κ2 = 10 − 2g, and with a) Δa = 55g, b) Δa = 5g.

Fig. 3a that is restricted by the condition of jΔa −Δb jbbΔa ; Δb , we set jΔa −Δb j≈Δb = 60g NN Δa = 5g in Fig. 3b, and we ﬁnd that in Fig. 3b, the area with larger degree of logarithmic negativity is larger than that in Fig. 3a. In the above, we focused our attention on the generation of entanglement between the two cavity modes in which the system is operated below the threshold, i.e., all the parameters which are chosen here ensured that all of the real parts of eigenvalues of the matrix M are negative. We can also show that when both the two Raman emission processes are open, the system can be operated above the threshold. Strictly speaking, Eq. (19) is a basic equation for the system which is obtained under the linear approximation and is not valid for the situation when the system is operated far above the threshold. Fortunately, when the system is operated above but very close to the threshold, Eq. (19) is also satisﬁed when the generated photon number is not very large. This means that Eq. (19) is valid not for a very long time period in which the generated photon number is not very large and the photon saturation effect can be ignored [17]. In Fig. 4, we plot the entanglement negativity and the mean number of

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modes strongly depends on the control parameters such as Rabi frequencies and the detunings of the pumping ﬁelds. To get the large degree of entanglement of the steady state, the difference between the two detunings of the atomic levels with the classical laser ﬁelds or the difference between the two Rabi frequencies must be larger than zero, and the high degree of entanglement is dependent on the proper values of the ratio between the two detunings and the ratio between the two Rabi frequencies. We also ﬁnd that the macroscopic entanglement can be generated when the system is operated above the threshold. Acknowledgment This work was supported by the Natural Science Foundation of China (Grant Nos. 10774117, 10604047, 60878004, and 60778021). þ and En (the inset) of two cavity Fig. 4. The photon number aþ 1 a1 + a2 a2 modes versus time gt. The solid line corresponds to the parameters of Ωa = 29.1g, Ωb = 25.3g, Δa = 5g, Δb = 60g, Δ = − 10g, g1 = g2 = g, γ = g, κ1 = κ2 = 10 − 3g, and the dashed line corresponds to the parameters of Ωa = 29g, Ωb = 18.3g, Δa = 22g, Δb = 45g, Δ = − 10g, g1 = g2 = g, γ = g, κ1 = κ2 = 10 − 3g.

þ photons aþ 1 a1 + a2 a2 versus time according to Eq. (24) when the system is operated above the threshold. The positive real parts of the eigenvalues of the matrix M are 0.0039 (for solid line) and 0.0184, 0.0059, 0.0059 (for dashed line), respectively. The ﬁgure shows that we can obtain entangled states with large mean number of photons, i.e. macroscopic entanglement. From the inset, we ﬁnd that the entanglement logarithmic negativity may have both steady values and maximum values in a certain small scaled time. Thus it is clear that macroscopic entanglement can be generated under the condition where neither of the two detunings Δa, Δb is not zero. 5. Conclusion In summary, we have considered the generation of entanglement from a four-level atomic system via the two Raman processes. It is easily realized experimentally that in our system neither of the detunings of the atomic levels with the classical laser ﬁelds and the two cavity modes is not to be zero necessarily. We ﬁnd that entanglement between the two cavity modes can be generated in both the time dependent case and steady case. Our results clearly show that when only one of the two Raman processes is operated, the system is approximately a three level optical nondegenerate parametric down conversion and entanglement can be generated under this case. When the two Raman processes are both operated with symmetric parameters, the system cannot generate entanglement. When the system is operated under other general conditions, the degree of entanglement for the cavity

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